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arXiv:2010.00538 · npj Quantum Information

Avoiding coherent errors with rotated concatenated stabilizer codes

Yingkai Ouyang

Published Jun 2, 2021• 26 citations
arXivHTMLPDF

Abstract

Abstract Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constant-excitation codes by code concatenation. Namely, by concatenating an [[n, k, d]] stabilizer outer code with dual-rail inner codes, we obtain a [[2n, k, d]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory.

DOI
10.1038/s41534-021-00429-8

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Abstract Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constant-excitation codes by code concatenation. Namely, by concatenating an [[n, k, d]] stabilizer outer code with dual-rail inner codes, we obtain a [[2n, k, d]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory.
Collective coherent (CC) errors are inevitable, as every physical qubit undergoes free evolution under its kinetic Hamiltonian. These errors can be more damaging than stochastic Pauli errors because they affect all qubits coherently, resulting in high-weight errors that standard quantum error-correcting (QEC) codes struggle to correct. In quantum memories and communication systems, especially when storage durations are long, CC errors often dominate over stochastic noise. Trapped-ion platforms, for example, exhibit strong CC errors with minimal stochastic Pauli components. In this work, we address the regime where immunity to CC errors, high code rate (due to limited qubit availability), and moderate distance (sufficient for correcting low-weight errors) are all essential. We construct a family of constant-excitation stabilizer codes with parameters $[[{2}^{r+1},{2}^{r}\ensuremath{-}(r+1),4]]$. The smallest instance, the [[8,1,4]] code, improves the code rate and error threshold of the best previously known CE code by factors of approximately two and four, respectively.
Collective coherent (CC) noise poses challenges for fault-tolerant error correction (FTEC), as it is not captured by conventional stochastic noise models. Constant-excitation (CE) codes are inherently immune to CC errors, but a fault-tolerant framework for operating these codes under circuit-level noise has not yet been established. Here, we develop an FTEC framework for CE CSS codes based on dual-rail concatenation. We show that conventional transversal CNOT gates violate the CE constraint and develop CE-preserving logical CNOT gates together with modified Shor- and Steane-type syndrome extraction schemes using zero-controlled NOT gates and CE-compatible ancilla states. We further develop an extended stabilizer simulation algorithm that tracks both stochastic and CC noise. Using this framework, we identify small distance-3 CE CSS codes demonstrate that the [[14,1,3]][[14,1,3]][[14,1,3]] code maintains robust performance under coherent noise. Our results establish a fault-tolerant framework for CE codes under circuit-level noise and demonstrate their potential for quantum processors affected by CC noise.
We formulate stabilizer-code error correction as a code- and recovery-dependent transformation of the noise channel. For a fixed recovery map, syndrome projection produces a block-diagonal logical channel whose diagonal Pauli weights follow from code-induced error classes, with no physical Pauli twirl; concatenation is then iterated redistribution, with the dominant logical error guiding the next code. The bit-flip and phase-flip three-qubit codes with standard recoveries supply a closed-form benchmark, and logically inequivalent recoveries appear as a design degree of freedom for shaping the channel presented to the next decoder. For amplitude damping, the three-qubit rules select C2(σ^y)C1(σ^x)C_2(\hatσ_y)C_1(\hatσ_x)C2​(σ^y​)C1​(σ^x​) and give fl=1−O(γ2)f_l=1-\mathcal O(γ^2)fl​=1−O(γ2), with an ideal-gate crossover at γc≈0.30γ_c\approx0.30γc​≈0.30 (f=1−1.72755γ2f=1-1.72755γ^2f=1−1.72755γ2) and a favorable gate-count comparison with the five-qubit code under the same assumptions. Correlated noise requires joint class statistics, whereas coherent noise retains intra-class coherences that make the same transformation usable in reverse as a tomographic fingerprint; there the companion work proves local identifiability from a rank certificate evaluated inside the parameter domain, classifies the sole permutation-symmetry ambiguity as structural at any chain length, and closes it with one two-qubit marginal channel. The construction remains an analytic benchmark, not a universal prescription for an inner code beneath a surface code.
Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. Stochastic errors in quantum systems occur randomly but coherent errors are more damaging since they can accumulate in a particular direction. We consider coherent noise acting transversally, giving rise to an effective error which is aZZZ-rotation on each qubit by some angle$\theta .Ratherthanaddresscoherentnoisethroughactiveerrorcorrection,weinvestigatepassivemitigationthroughdecoherencefreesubspaces.Inthelanguageofstabilizercodes,werequirethenoisetopreservethecodespace,andtoacttrivially(asthelogicalidentityoperator)ontheprotectedinformation.Thus,wedevelopnecessaryandsufficientconditionsforalltransversal. Rather than address coherent noise through active error correction, we investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversal.Ratherthanaddresscoherentnoisethroughactiveerrorcorrection,weinvestigatepassivemitigationthroughdecoherencefreesubspaces.Inthelanguageofstabilizercodes,werequirethenoisetopreservethecodespace,andtoacttrivially(asthelogicalidentityoperator)ontheprotectedinformation.Thus,wedevelopnecessaryandsufficientconditionsforalltransversalZ$-rotations to preserve the code space of a stabilizer code. These conditions require the weight-2 Z2~Z2 Z-stabilizers to cover all the qubits that are in the support of theXXX-component of some stabilizer. Furthermore, the weight-2 Z2~Z2 Z-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the sizes of these components, we are able to construct a large family of QECC codes oblivious to coherent noise, one that includes the[[4L2,1,2L]][[4L^{2}, 1, 2L]][[4L2,1,2L]]Shor codes. The Shor codes are examples of constant excitation codes, where logical qubits are encoded as a code state that is a sum of physical states indexed by binary vectors with the same weight. Constant excitation codes are oblivious to coherent noise since a transversalZZZ-rotation acts as a global phase. We prove that a CSS code is oblivious to coherent noise if and only if it is a constant excitation code, and that if the code is error-detecting, then the (constant) weights in different cosets of theXXX-stabilizers are identical.
Entanglement-assisted (EA) stabilizer codes enhance the rate of error correction in relation to codes with no pre-shared entanglement. Meanwhile, bosonic error-correcting codes, such as the Gottesman-Kitaev-Preskill (GKP) code, can be concatenated with qubit stabilizer codes to significantly reduce the logical failure probability of those stabilizer codes. First, we combine the above two concepts to propose an EA version of the qubit-into-oscillators concatenated code that chains an EA-stabilizer (outer) code with a GKP (inner) code. As an example we present a three-qubit EA-repetition concatenated with a GKP code. Second, we propose an EA version of the non-Gaussian oscillator-into-oscillators concatenated code that chains a GKP (outer) code with an EA-stabilizer (inner) code. As an example we present a GKP code concatenated with a three-qubit EA repetition code that uses two maximally entangled modes (emodes) and suppresses the variances of both position and momentum quadrature errors of a data mode. Furthermore, we generalize the latter example to a family of GKP code concatenated with a nnn-qubit EA repetition code that uses n−1{n-1}n−1 emodes and suppresses the variances of both position and momentum quadrature errors of a data mode by a factor 1/n{1/n}1/n.
We introduce a family of high-rate amplitude-damping (AD) Shor codes, designed to effectively correct AD errors while maintaining immunity to collective coherent (CC) errors. The proposed [[(w+1)(w+K),K]][[(w+1)(w+K),K]][[(w+1)(w+K),K]] AD codes can approximately correct up to www AD errors, with flexible parameters (w,K)(w,K)(w,K), and we provide a rigorous proof that these codes satisfy the approximate quantum error-correction conditions. These codes leverage structured stabilizer measurements, enabling efficient detection of AD errors using only local operations and ancillary qubits. We further construct a family of CC-AD Shor codes by concatenating these AD codes with the dual-rail code.